Sine-Generated Meander Patterns

Sine-generated plane curves exhibit a wide variety of looping patterns. They are constructed in natural coordinates in which the tangential angle &theta of the curve is a function of the arc length s. We define sine-generated curves as the family of curves in which the function θ(s) may be written as a sum of N sine waves:

Each sine term has three independent parameters: the amplitude Ai, wavelength λi and phase constant φi. Once the θ(s) function is specified, the x and y coordinates of the function are obtained by integration:

Single-Frequency Curves

When only a single sine term is included in the series (i.e. when N = 1), the shape of the curve is determined soley by the amplitude A. The applet below (written in processing) shows a single-frequency sine-generated curve. Moving the cursor horizontally across the graph adjusts the amplitude.

Multi-Frequency Curves

The multi-frequency sine-generated applet allows the user to control the amplitude and wavelength of up to three sine terms.

When the applet is first launched, the curve defaults to two sine components. The cursor x position controls the wavelength and the cursor y position controls the amplitude of the second sine term.

Clicking on the Greek letters at the bottom of the window resets the parameter values to preset values, giving a hint of the diverse patterns that this family of curves can produce.

The "i" box at the bottom right brings up a control panel. Typing "i" on the keyboard does the same.

Typing "w" colors the curve according tot he winding number.

The LEFT and RIGHT arrow keys act as slow motion adjustment for the parameter controlled by the x cursor motion (amplitude by default).

The UP and DOWN arrow keys act as slow motion adjustment for the parameter controlled by the y cursor motion (wavelength by default).

Pressing SPACE disables the cursor control of the parameters (to regain control, you must hit the "i" button to expand the full control panel).

To zoom in and out, type "+" and "–" .

Expanded Control Window:

The full control window may be expanded or hidden by clicking on the info "i" box at the bottom right of the window. Typing "i" does the same.

The full control panel shows the values for the amplitudes and wavelengths for each of the sine components. Slider bars allow the user to adjust these parameters directly.

The X and Y buttons to the right of the slider bar allow the user to select which parameter is adjusted when the cursor is over the plot window. For example, clicking "X" for A1 and "Y" for A2 would allow the user to adjust those two parameters by moving the curson over the plot window. The X and Y buttons may be toggled to the off position as well to allow adjustment of a single parameter at a time.

The Length of Curve slider changes the total arc length of the curve. WARNING!! Large values will cause the applet to SLOW way down. Use with care.

The zoom slider changes the scale of the plot, effectively zooming in and out.

The Winding Number button colors the plot by the winding number. For complex curves, the rendering can take several seconds.